TY  - JOUR
A1  - Becker, Christian
A1  - Schenkel, Alexander
A1  - Szabo, Richard J.
T1  - Differential cohomology and locally covariant quantum field theory
JF  - Reviews in Mathematical Physics
N2  - We study differential cohomology on categories of globally hyperbolic Lorentzian manifolds. The Lorentzian metric allows us to define a natural transformation whose kernel generalizes Maxwell's equations and fits into a restriction of the fundamental exact sequences of differential cohomology. We consider smooth Pontryagin duals of differential cohomology groups, which are subgroups of the character groups. We prove that these groups fit into smooth duals of the fundamental exact sequences of differential cohomology and equip them with a natural presymplectic structure derived from a generalized Maxwell Lagrangian. The resulting presymplectic Abelian groups are quantized using the CCR-functor, which yields a covariant functor from our categories of globally hyperbolic Lorentzian manifolds to the category of C∗-algebras. We prove that this functor satisfies the causality and time-slice axioms of locally covariant quantum field theory, but that it violates the locality axiom. We show that this violation is precisely due to the fact that our functor has topological subfunctors describing the Pontryagin duals of certain singular cohomology groups. As a byproduct, we develop a Fréchet–Lie group structure on differential cohomology groups.
KW  - Algebraic quantum field theory
KW  - generalized Abelian gauge theory
KW  - differential cohomology
Y1  - 2017
U6  - https://doi.org/10.1142/S0129055X17500039
SN  - 0129-055X
SN  - 1793-6659
VL  - 29
IS  - 1
PB  - World Scientific
CY  - Singapore
ER  - 
TY  - JOUR
A1  - Becker, Christian
A1  - Benini, Marco
A1  - Schenkel, Alexander
A1  - Szabo, Richard J.
T1  - Cheeger-Simons differential characters with compact support and Pontryagin duality
JF  - Communications in analysis and geometry
N2  - By adapting the Cheeger-Simons approach to differential cohomology, we establish a notion of differential cohomology with compact support. We show that it is functorial with respect to open embeddings and that it fits into a natural diagram of exact sequences which compare it to compactly supported singular cohomology and differential forms with compact support, in full analogy to ordinary differential cohomology. We prove an excision theorem for differential cohomology using a suitable relative version. Furthermore, we use our model to give an independent proof of Pontryagin duality for differential cohomology recovering a result of [Harvey, Lawson, Zweck - Amer. J. Math. 125 (2003), 791]: On any oriented manifold, ordinary differential cohomology is isomorphic to the smooth Pontryagin dual of compactly supported differential cohomology. For manifolds of finite-type, a similar result is obtained interchanging ordinary with compactly supported differential cohomology.
Y1  - 2019
U6  - https://doi.org/10.4310/CAG.2019.v27.n7.a2
SN  - 1019-8385
SN  - 1944-9992
VL  - 27
IS  - 7
SP  - 1473
EP  - 1522
PB  - International Press of Boston
CY  - Somerville
ER  -